Positive solutions for a weakly coupled nonlinear Schrödinger system

Abstract

Existence of a nontrivial solution is established, via variational methods, for a system of weakly coupled nonlinear Schrödinger equations. The main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero. Generalizations for nonautonomous systems are considered.

Gain in regularity for a coupled nonlinear Schrödinger system

“…This operator has the remarkable property that it commutes with the operator 8) which allows us to get the estimates (1.9)-(1.10) of the Main Theorem. Throughout this paper c is a generic constant, not necessarily the same at each occasion (it will change from line to line), which depends in an increasing way on the indicated quantities.…”

“…The Cauchy problem for the system (1.1)-(1.4) was firstly studied by E. S. P. Siqueira [13,14] for initial data u 0 ∈ H 1 (R) and v 0 ∈ H 1 (R), then the solution u ∈ C(R : H 1 (R)) ∩ C 1 (R : H −1 (R)) and v ∈ C(R : H 1 (R)) ∩ C 1 (R : H −1 (R)), using the techniques developed in [1,2]. This Schrödinger system has been extensively studied for many authors [6,8,9,10,11] and references therein. An evolution equation enjoys a gain of regularity if their solutions are smoother for t > 0 than its initial data.…”

Positive Radial Solutions for a Class of Elliptic Systems Concentrating on Spheres With Potential Decay

“…[1,20,21,22]), Hartree-Fock theory for double condensate [25]. See [31] and [37] for more applications in physical and chemical phenomenas.…”

Uniqueness of Positive Solutions for a Nonlinear Elliptic System

“…The system (5) arises in many physical problems, especially in the Hartree-Fock theory and nonlinear optics. We refer to [1,3,6,[8][9][10]13,14] and references therein.…”